Back to Non-Intersecting Page.
This page was begun on 16 April 2015.
This new page is to record solutions to the problem of constructing knight's paths of maximum length without self-intersection and showing binary symmetry on square boards of all sizes. It is assumed that the board also participates in the symmetry. Thus symmetric paths offset from the centre, or with unused outer ranks and files are not included. This page is a supplement to the more general one on open and closed paths without the symmetry restriction. There is also a separate page for paths showing higher symmetry.
Symmetric open tours require a board with an odd side.
The tours of this type are all oriented with the central two moves sloping gently up to the right.
So that the efficiency of open and closed tours of the same length shall be the same it is convenient to define e = L/n²
rather than in terms of the number of cells, C, as was done previously.
3×3 There is no tour on the 3×3 board, although to preserve the relation C = L + 1 in an open tour one may wish to consider that a knight placed on the centre cell constitutes a tour of length 0, using one cell.
4×4 There are no solutions with purely binary symmetry on the 4×4 board, though there are tours with higher symmetry, using 4 moves, giving an effficiency of 4/16 = 1/4 = 25%.
5×5 8-move tours open and closed (32%).
6×6 12-move tours (33.3%). No symmetric open tour is possible on an even-sided square, since there has to be a central cell.
7×7 Three 16-move closed tours (24 is possible with higher symmetry). Two 24-move open tours. The first of these is by Knuth (1969). I'm not sure if he also gave the second solution, or if that is a new discovery of my own. I also found 10 solutions with 22 moves.
8×8 24-move tour. (28 is possible with higher symmetry.)
9×9 32-move tours (40 is possible with higher symmetry).
10×10 52-move tour (52%) by George Jelliss (15 April 2015).
14×14 112-move tour by Bernard Lemaire and a variant of this by George Jelliss.
In both, further variation is possible by folding in or out of acute-angled Vs along various diagonals.
This is the first case where the binary symmetry solution is longer than for higher symmetry, where ony 108 has been achieved.
15×15. open, 158 moves (159 cells), symmetric, by Robin Merson (1990). This and the related tour of side 23 are the only symmetric solutions that Robin produced.
16×16 164-move tour (64%) by George Jelliss (18 March 2015). This equals the higher symmetry case.
23×23, open, 414 moves (415 cells), symmetric, by Robin Merson (1990) by VW extension from 15×15 example.
CLOSED TOURS. The table gives the maximum number of moves achieved for each board size under the binary symmetry condition.
n = number of cells in side of square. L (length) = number of moves used. e (efficiency) = L/n²
| side | L | e | side | L | e | side | L | e |
| 3 | 0 | 0 | 13 | - | - | 23 | - | - |
| 4 | 0^ | 0 | 14 | 112 | 57.14% | 24 | - | - |
| 5 | 8 | 32% | 15 | - | - | 25 | - | - |
| 6 | 12 | 33.33% | 16 | 164 | 64.06% | 26 | - | - |
| 7 | 16 | 32.65% | 17 | - | - | 27 | - | - |
| 8 | 24 | 37.5% | 18 | - | - | 28 | - | - |
| 9 | 32 | 39.51% | 19 | - | - | 29 | - | - |
| 10 | - | - | 20 | - | - | 30 | - | - |
| 11 | - | - | 21 | - | - | 31 | - | - |
| 12 | - | - | 22 | - | - | 32 | - | - |
OPEN TOURS. The table gives the maximum number of moves achieved for each board size (necessarily odd).
n = number of cells in side of square. L (length) = number of moves used. e (efficiency) = L/n²
| side | L | e | side | L | e | side | L | e |
| 3 | 0 | 0 | 13 | - | - | 23 | 414 | 78.26% |
| 5 | 8 | 32% | 15 | 158 | 70.22% | 25 | - | - |
| 7 | 24 | 48.98% | 17 | - | - | 27 | - | - |
| 9 | - | - | 19 | - | - | 29 | - | - |
| 11 | - | - | 21 | - | - | 31 | - | - |